There was plenty of concern with witchcraft in a variety of Christian groups and denominations, not just the Puritans, at the same time and in the preceding centuries. There are Church records, both in canon law and in correspondence between various levels of the Catholic clerical hierarchy, that address witchcraft and sorcery and how to combat it going back to at least 11th Century, but the widespread interest seems to have peaked right around the time of the Reformation. This may be connected to the general sense of religious upheaval going on at the time, and was almost certainly influenced by the development of the printed word. Turns out the supernatural and occult have been popular themes in publishing for a long while.

The Malleus Maleficarum from the 1480s is the best known and probably most influential witch-hunting guidebook from this period and is very useful in terms of appreciating and understanding the worldview of the people of the time. From the 15th to 17th Centuries, there were witch-hunts all over Northern and Western Eruope-France, Germany, Norway, Sweden, England all saw with-hunts and witch-trials at various points. The American Puritan witch-hunting that we're most commonly familiar with now actually is just a small piece of a much greater trend towards concern for the occult in this period in Western European Christianity.

In terms of where these people believe the witchcraft originated, the answers are going to vary to some degree I'm sure, but generally the Christian understanding has been that witchcraft and sorcery comes from the devil or demons, and frequently there was some sort of sexualized component to this arrangement. Witches were thought to essentially engage in demonic orgies as part of their initiations and covens.

It might be easier to think of witchcraft as being connected to the ideas of demonic possession and exorcism in terms of how it was approached on a religious level. The understand was that either the devil himself or his demons would lead (usually) a woman into temptation by capitalizing on weakness, and then they would enter into some form of contract giving the devil control or authority over the witch in exchange for magical powers.